Starship Weapon System Maintenance, and the rest of the skills under Starship Engineering and the likes, do they improve your weapon system abilities as a general (i.e. faster recharge rate, more damage a.s.o.) or do they just improve the abilities like they say (Emergency Power to Weapons, Directed Energy Modulation a.s.o.)?

I'm fairly certain that talent directly improves your weapons fire, since there are dedicated talents for those at later ranks. It may effect the GCD if your weapons, but that's neither here nor there. It probably does just what the talent says it does- makes your bridge officers more effective.

The skills for the weapon types (Starship Energy Weapons, Starship Projectile Weapons, etc.) improve damage & accuracy when using those types of weapons. The maintenance skills improve the powers specifically stated. They may also improve some other stuff, but I'd go by the explicitly listed information first.

In the tier 3 skill list the engineering skills that say Efficiency directly improve your power levels in those areas. It's a great idea to max out all 4 of those.
In the tier 2 skill list though, those skill points only affect the abilities that correspond with them. So only put points in if you are using or plan to use abilities that will benefit from them.